Chelsea had hoped Drogba would play at least one game before leaving for international duty on January 7.
But his confession could now spark a club v country row, with the Blues unwilling to risk him if he is not fit.
Drogba said: "No player would go under the knife at this crucial time if it was not serious.
"I can't say really when I'll be back.
"There's a possibility of missing the African Nations Cup."
Ivory Coast boss Uli Stieleke insisted: "Our training camp starts on January 7 and Didier will be there."